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Page 14 - In any triangle, the product of two sides is equal to the product of the segments of the third side formed by the bisector of the opposite angle plus the square of the bisector.
Page 10 - If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts.
Page 17 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sidef.
Page 23 - ... distance between the centres of the inscribed circle and of the circle through the middle points of sides has been proved to be exactly the difference between their radii ; and the same argument applies to any of the four circles which touch the three sides of the given triangle ; hence (5) The circle which passes through the middle points of the sides of a triangle touches the four circles which touch the three sides. This theorem was new both to Dr Hart and myself,* but I have lately learned...
Page 8 - THEOREM. 35. Of two oblique lines drawn from the same point to the same straight line, that is the greater which cuts off upon the line the greater distance from the perpendicular. Let PC be the perpendicular from P to AB, and suppose CE > CD; thenPU> PD. For, produce PC to P', making CP
Page 15 - Two triangles are similar if an angle of one equals an angle of the other and the sides including these angles are proportional.
Page 18 - Two diagonals of a regular pentagon, not drawn from a common vertex, divide each other in extreme and mean ratio.
Page 31 - The surface of a sphere is equal to the product of its diameter by the circumference of a great circle.